Example: Using Gauß-Jordan algorithm for matrix inversion

2.10.2 Example: Using Gauß-Jordan algorithm for matrix inversion

For the Gauß-Jordan algorithm for finding the inverse of a matrix, which is the most commonly used technique, we will give here just one example:

\[\begin{array}{ccc} \mbox{\bf Matrix}& & \mbox{\bf Identity matrix}\\ \left(\begin{array}{ccc}1&0&1\\ 2&1&0\\ 3&0&1\end{array}\right)& (III - I) / 2 \rightarrow I&\left(\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right)\\& & \\ \left(\begin{array}{ccc}1&0&0\\ 2&1&0\\ 3&0&1\end{array}\right)& II/2 - I \rightarrow I &\left(\begin{array}{ccc}-\frac{1}{2}&0&+\frac{1}{2}\\ 0&1&0\\ 0&0&1\end{array}\right)\\& & \\ \left(\begin{array}{ccc}1&0&0\\ 0&1&0\\ 3&0&1\end{array}\right)& II/2 - I \rightarrow II &\left(\begin{array}{ccc}-\frac{1}{2}&0&+\frac{1}{2}\\ 1&1&-1\\ 0&0&1\end{array}\right)\\& & \\ \left(\begin{array}{ccc}1&0&0\\ 0&1&0\\ 3&0&1\end{array}\right)& III/3 - I \rightarrow III &\left(\begin{array}{ccc}-\frac{1}{2}&0&+\frac{1}{2}\\ 1&1&-1\\ 0&0&1\end{array}\right)\\& & \\ \left(\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right)& &\left(\begin{array}{ccc}-\frac{1}{2}&0&+\frac{1}{2}\\ 1&1&-1\\ \frac{3}{2}&0&-\frac{1}{2}\end{array}\right)\\ \mbox{\bf Identity Matrix}& & \mbox{\bf Inverse matrix} \end{array}\]

With frame

Regular matrix, inverse matrix